Brown the meat in a large soup pot over medium heat. Halfway through cooking the meat, add the onions and continue to cook until meat is completely brown and onions are soft - about 7-10 minutes.
Add remaining veggies to the pot: celery, carrots, radishes, and cabbage. Stir to mix the meat & veggies together.
Pour in diced tomatoes, beef broth, and tomato sauce in soup pot & stir.
Bring to a simmer over medium heat. Once the soup is at a simmer, reduce heat to medium-low, cover, and allow to simmer for at least an hour.
Serve with shredded mozzarella or Italian-style cheese on top.